2.3 Apply Deductive Reasoning Objectives: 1.To recognize deductive reasoning and use it to arrive at a true conclusion.

History When the architects designed this school building, they were approached by an ancient secret society whose members make up numerous Texas dignitaries. They convinced the architects to add several secret passages and hidden conference rooms to their design plans.

History Sometimes when I stay after school late into the evening grading papers, planning lessons, and contacting parents, I hear strange and inauspicious sounds emanating from behind one of my walls.

Conspiracy? Thus, it is my conjecture that one of the secret society's hidden passages lies between the walls of Room 701 and 703. This is a bold and perhaps conspiratorial conjecture, but I am confident that it is true. (You should hear the sounds--Oh, my!)

Stop Making Fun of Me! I have told few people of my theory, and they unanimously dismiss my conviction with ridicule. (Then they ask me if I frequently watch re-runs of the X- Files with the notion that the story lines are largely nonfiction!)

Redemption To convince the skeptics and to redeem my reputation, I need absolute and conclusive proof that there exists a hidden passage between these classrooms.

Proof The Principle of Laplace: The weight of evidence for an extraordinary claim must be proportional to its strangeness.

In Other Words… “Extraordinary claims require extraordinary evidence.” -Carl Sagan

Example 1 In your group, come up with a nondestructive method for proving or disproving the extraordinary claim that there’s a secret tunnel between 701 and 703.

Example 2 In the Sudoku puzzle shown, what number must be written in the blue box? Why? ?

Deductive Reasoning The process of demonstrating that if certain statements are accepted as true, then other statements can be shown to follow from them.

Deductive Reasoning premisesassumptions The “accepted” statements are sometimes premises or assumptions, and all deductive arguments must have them. Deductive reasoning uses logical inference to build on these assumptions. Unlike inductive reasoning, deductive reasoning will always lead to the truth as long as the assumptions are true.

Example 2 All humans have skeletons is a reasonable assumption. So, since Mrs. DeZeeuw is a human, what must be true about her?

Deductive Reasoning

Inductive vs. Deductive 1.We use inductive reasoning to investigate and discover things about our world. 2.Since the conjectures we make using our inductive reasoning are based on our fallible observation skills, we can be wrong. 3.We can search for a counterexample to disprove our conjectures. 4.In mathematics, we use our deductive reasoning to prove our conjectures beyond all uncertainty.

Example 4 Follow along as with an excerpt from The Adventure of the Dancing Men by Sir Arthur Conan Doyle, paying particular attention to the deductions made by Sherlock Holmes. Click here

Flavors of Deductive Reasoning Deductive reasoning comes in a variety of flavors: 1.Law of Detachment 2.Denying the Consequent 3.Law of Syllogism

Law of Detachment SymbolsExample If Watson had chalk on his fingers, then he had been playing billiards. Watson had chalk between his fingers upon returning from the club. Therefore Watson had been playing billiards.

Denying the Consequent SymbolsExample If Watson wished to invest his money in S. African securities with Thurston, then he would have had his check book when playing billiards with Thurston. Watson did not have his checkbook when he played billiards with Thurston. Therefore Watson did not wish to invest his money in S. African securities with Thurston.

Law of Syllogism SymbolsExample If I eat pizza after midnight, then I will have nightmares. If I have nightmares, then I will get very little sleep. Therefore, if I eat pizza after midnight, then I will get very little sleep.

Example 5 Use one of the laws of deductive reasoning to make a valid conclusion. If two segments have the same length, then they are congruent. You know that BC = XY. Law of Detachment

Example 6 Use one of the laws of deductive reasoning to make a valid conclusion. If x > 5, then x 2 > 25. If x 2 > 25, then x 2 > 20. Therefore, x>5 so x 2 >20 Law of Syllogism

Example 7 Use one of the laws of deductive reasoning to make a valid conclusion. If a polygon is regular, then it is both equilateral and equiangular. Pentagon ABCDE is not equilateral or equiangular. Denying the Consequent